Solving Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.

Example

Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her $4 and each large photo costs her $10. She doesn’t want to spend more than $200 on photos to display.

Write a system of inequalities to model this situation.
Graph the system.
Could she display 15 small and 5 large photos?
Could she display 3 large and 22 small photos?

Solution

Let $x=$ the number of small photos.
$\phantom{\rule{2.5em}{0ex}}y=$ the number of large photos

To find the system of inequalities, translate the information.
$\begin{array}{ccccc}& & & & \text{She wants to have at least 25 photos.}\hfill \\ & & & & \text{The number of small plus the number of large should be at least 25.}\hfill \\ & & & & \hfill x+y\ge 25\hfill \\ & & & & \text{\$4 for each small and \$10 for each large must be no more than \$200}\hfill \\ & & & & \hfill 4x+10y\le 200\hfill \end{array}$

We have our system of inequalities. $\begin{array}{c}x+y\ge 25\hfill \\ 4x+10y\le 200\hfill \end{array}$

To graph $x+y\ge 25$, graph x + y = 25 as a solid line.

Choose (0, 0) as a test point. Since it does not make the inequality true, shade the side that does not include the point (0, 0) red.

To graph $4x+10y\le 200$, graph 4x + 10y = 200 as a solid line.

Choose (0, 0) as a test point. Since it does not make the inequality true, shade the side that includes the point (0, 0) blue.

The solution of the system is the region of the graph that is double shaded and so is shaded darker.

To determine if 10 small and 20 large photos would work, we see if the point (10, 20) is in the solution region. It is not. Christy would not display 10 small and 20 large photos.
To determine if 20 small and 10 large photos would work, we see if the point (20, 10) is in the solution region. It is. Christy could choose to display 20 small and 10 large photos.

Notice that we could also test the possible solutions by substituting the values into each inequality.

Example

Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than $5. At the hamburger restaurant near his university, each hamburger has 240 calories and costs $1.40. Each cookie has 160 calories and costs $0.50.

(a) Write a system of inequalities to model this situation.
(b) Graph the system.
(c) Could he eat 3 hamburgers and 1 cookie?
(d) Could he eat 2 hamburgers and 4 cookies?

Solution

(a) Let $h=$ the number of hamburgers.

$\phantom{\rule{2.55em}{0ex}}c=$ the number of cookies

To find the system of inequalities, translate the information.

The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.

$240h+160c\ge 800$

The amount spent on hamburgers at $1.40 each, plus the amount spent on cookies at $0.50 each must be no more than $5.00.

$1.40h+0.50c\le 5$

We have our system of inequalities. $\phantom{\rule{3em}{0ex}}\begin{array}{c}240h+160c\ge 800\hfill \\ 1.40h+0.50c\le 5\hfill \end{array}$

(b)

To graph $240h+160c\ge 800$ graph $240h+160c=800$ as a solid line.

Choose (0, 0) as a test point. it does not make the inequality true.

So, shade (red) the side that does not include the point (0, 0).

To graph $1.40h+0.50c\le 5$, graph $1.40h+0.50c=5$ as a solid line.

Choose (0,0) as a test point. It makes the inequality true. So, shade (blue) the side that includes the point.

The solution of the system is the region of the graph that is double shaded and so is shaded darker.To determine if 3 hamburgers and 2 cookies would meet Omar’s criteria, we see if the point (3, 1) is in the solution region. It is. He might choose to eat 3 hamburgers and 2 cookies.

To determine if 2 hamburgers and 4 cookies would meet Omar’s criteria, we see if the point (2, 4) is in the solution region. It is. He might choose to eat 2 hamburgers and 4 cookies.We could also test the possible solutions by substituting the values into each inequality.

Access these online resources for additional instruction and practice with graphing systems of linear inequalities.

Solving Applications of Systems of Inequalities